let L be non empty join-commutative meet-commutative distributive join-idempotent upper-bounded' lower-bounded' distributive' complemented' LattStr ; :: thesis: L is complemented

for b being Element of L ex a being Element of L st a is_a_complement_of b

for b being Element of L ex a being Element of L st a is_a_complement_of b

proof

hence
L is complemented
; :: thesis: verum
let b be Element of L; :: thesis: ex a being Element of L st a is_a_complement_of b

consider a being Element of L such that

A1: a is_a_complement'_of b by Def7;

take a ; :: thesis: a is_a_complement_of b

A2: b "/\" a = Bot' L by A1;

b "\/" a = Top' L by A1;

hence ( a "\/" b = Top L & b "\/" a = Top L & a "/\" b = Bottom L & b "/\" a = Bottom L ) by A2, Th18, Th19; :: according to LATTICES:def 18 :: thesis: verum

end;consider a being Element of L such that

A1: a is_a_complement'_of b by Def7;

take a ; :: thesis: a is_a_complement_of b

A2: b "/\" a = Bot' L by A1;

b "\/" a = Top' L by A1;

hence ( a "\/" b = Top L & b "\/" a = Top L & a "/\" b = Bottom L & b "/\" a = Bottom L ) by A2, Th18, Th19; :: according to LATTICES:def 18 :: thesis: verum